Model predictive control of induction motor based on amplitude–phase motion equation

: Tedious adjustment of the weighting factor restricts the wide application of conventional model predictive torque control (MPTC) for induction motors (IMs) in practice. To solve this problem, this study introduces a new variable R (reactive torque) to control the flux of IM, which is the dual of torque T , with radial orientation and in relation to reactive power. Then, a novel MPTC strategy based on variable R and torque T is proposed and named RT-MPC. Compared with conventional MPTC, RT-MPC uses a pair of variables with the same dimensions and time scales instead of the torque and flux amplitude in MPTC. This approach eliminates the weighting factor of the cost function, unifies the time scales of control variables, and avoids prediction of rotor flux at the ( k + 2) instant. Simulation and experimental results show that RT-MPC can avoid the weighting factor, reduce switching frequency, and reduce control complexity. RT-MPC can also maintain excellent dynamic and steady- state performance similar to conventional MPTC. Hence, the proposed method is expected to be more practical.


Introduction
Model predictive control (MPC) originated in industrial process control and has presented great advantages in dealing with the complex constrained optimisation problems of non-linear systems. MPC strategies have received wide attention in research communities due to merits of high flexibility, a simple concept, and easy inclusion of non-linearities and constraints [1][2][3][4][5]. In recent years, MPC has been applied in the field of power electronic control [6][7][8][9][10]. Some scholars have also tried to introduce the approach into direct torque control (DTC) with some success.
In conventional model predictive torque control (MPTC), the cost function usually consists of torque and flux magnitude errors, which differ in dimension and time scale, and thus lead to a tedious weighting factor design and unsatisfactory control [11][12][13]. However, tuning of the weighting factor is not easy due to the lack of a theoretical design procedure [14]. A method of online optimisation of the weighting factor for induction motors (IMs) was proposed in [15], in which the weighting factor was optimised by minimising the torque ripple. This method is feasible, but the calculation is highly complex. In [14], the optimal voltage vector was selected by calculating the torque and flux cost functions for all voltage vectors and then sorting the two cost functions. Although this method does not require setting a weighting factor, it does require online sorting, and the algorithm is complicated. A three-vector method was used to eliminate weighting factors in [16], but the method is complex, and calculating the vector duration is tedious. Recently, a model predictive flux control scheme was proposed in [17] to avoid the weighting factor. However, this method requires division calculation, and the calculation is again complex.
To eliminate the weighting factor and unify the time scales of reference variables in the cost function of conventional MPTC, this paper studies the dynamic characteristics of flux and torque based on the mathematical model [18,19] of IM. Fig. 1a in [18] and Fig. 1b in [19] clarify that the 'tangential force' f T and 'radial force' f R are critical control variables in electromechanical systems, as shown in Fig. 1. The essence of f T and f R is that active power adjusts the phase, whereas reactive power controls the amplitude. Moreover, Fig. 1b indicates that torque T is related to f T and corresponds to active power; although related to f R , the other key variable corresponding to reactive power is not given. Fig. 1a points out that the radial motion is relatively slow compared with the tangential motion. In other words, the change of amplitude is slower than that of phase. The variable R proposed in this paper makes up for the lack of 'the other key variable', and the idea of separating control according to different time scales fully embodies the conclusion of [18].
Electromagnetic torque T can be expressed by the external product of stator flux and rotor flux vectors, with a tangential orientation and in relation to active power. To the best of the author's knowledge, the electromagnetic torque T has been studied extensively, but research on its dual variable is relatively rare. In this paper, we apply the ghost operator g [20] to torque T to obtain its dual variable R, which is defined by the inner product of the stator flux and rotor flux vectors. According to the g operator theory, reactive torque R is in the radial orientation and related to reactive power, hence the name. In an IM system, instantaneous reactive power and active power correspond precisely with excitation and torque, respectively. Thus, variable R is closely related to motor flux dynamics and has been employed to prove the stability of DTC in [21][22][23].
The expressions of R and T are nearly the same except for the sine and cosine of the angle difference, which can reveal the physical meaning of reactive power and active power. Moreover, the following phenomena have been discovered in the mechanical system:

Fig. 1 Control model of IM
(a) Mechanical model in [18], (b) Mechanical analogy in [19] IET Power Electron., 2019, Vol. 12 Iss. 9, pp. 2400-2406 This is an open access article published by the IET under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0/) (1) Variable R and torque T are direct current signals, which can be easily observed and used to fully characterise the flux dynamics and torque dynamics of IM, respectively; (2) R and T have fast time characteristics, but speed and flux have slow time characteristics [24]; (3) The en-stabilising (i.e. other variables maintain the stability of the studied variable when the system is disturbed) [24] torque is predominated by positive feedback non-linear automatic gain control; (4) The self-stabilising (i.e. able to maintain its own stability) [24] flux magnitude is predominated by negative feedback non-linear automatic gain control. Therefore, this paper defines a new cost function consisting of tracking errors of R and T to select the optimal voltage vector. As a result, the weighting factor is naturally eliminated, and the time scales of reference variables become unified.
The remainder of this paper is organised as follows. Section II presents the torque predictive model of IM. In Section III, derivation of the new control scheme (RT-MPC) for IM is discussed. Validation results are reported in Sections IV and V, including MATLAB/Simulink simulation tests and experiments using rapid-control-prototyping controller dSPACE DS1104. Finally, conclusions are given in Section VI.

Conventional MPTC
A conventional MPTC scheme includes three main steps: (1) variable estimation, (2) prediction of future plant behaviour, and (3) cost function optimisation. The MPTC scheme is shown in Fig. 2. The implementation of MPTC depends heavily on the system model.

Electromechanical system
The mathematical model of IM can be expressed in a stationary frame as follows [14]: where u s and i s are the stator voltage and current vectors; ψ s and ψ r are the stator and rotor flux vectors; L s , L r , and L m are the stator, rotor, and mutual inductance, respectively; R s and R r are the stator and rotor resistances; ω is the electrical speed; T is the electromagnetic torque; p is the number of pole pairs, λ = 1/(L s L r − L m 2 ); and ⊗ is the external product operator. In MPTC, stator flux and rotor flux can be easily estimated based on measurements of the stator current and rotor speed. Conventionally, the rotor current model of IM [25] can be used to estimate rotor flux. Then, the simple correlation between stator and rotor flux can be applied to estimate stator flux.

Stator flux and electric torque prediction
From the IM model described above, the stator current can be written as follows: Where By using the forward Euler method to discretise (1) and (6), the stator flux and stator current at the (k + 1) instant can be predicted as where T s is the sampling time, u s (k) is the applied voltage vector, With ψ s (k + 1) and i s (k + 1), the rotor flux at the (k + 1) instant can be calculated as The electromagnetic torque at the (k + 1) instant can be calculated as

Optimisation of cost function
In MPTC [6], the predefined cost function consisting of the torque and flux errors is used to select the best voltage vector: where the torque reference T* is generated externally by a PI-speed controller, ψ s * is the stator flux reference, and λ ψ is the weighting factor.
Torque and flux have different dimensions and time scales, but the cost function is built by a linear combination of the two. Hence, the weighting factor λ ψ is needed to adjust the importance of the flux over the torque control. The weighting factor for stator flux must be tuned carefully to obtain satisfactory performance at different operating points [26]. However, at present, design of the weighting factor is based primarily on analytical [15] and empirical [27] procedures. Naturally, the adjustment of λ ψ is tedious and highly affects control system performance.

Proposed RT-MPC model
The structure of the proposed RT-MPC is shown in Fig. 3 and mainly consists of the following parts: stator flux and stator current estimation, prediction of variable R and torque T with delay compensation, and cost function minimisation. Additionally, the references of variable R and torque T are obtained by outer PI-flux and PI-speed controllers, respectively. This is a hybrid design of field-oriented control (FOC), which has a separation property of the torque with fast time-scale dynamics and flux with slow time- scale dynamics; and DTC, which directly controls torque without the current loop. Finally, the field-weakening operation is not considered in this paper, hence we assume the magnitude of the stator flux to be constant. The details of RT-MPC are discussed in more detail in the following sections.

Amplitude-phase motion equation of IM
The torque formula can be rewritten as follows: Where θ is the angle between the stator and rotor flux vectors. T is computed from the z component of the vector product of two state variables, as shown in Fig. 4. Note that in the rest of this paper, T only requires numerical calculations, so the direction z is omitted. The torque formula expressed by the stator and rotor flux vectors in DTC shows the rotation motion of the stator flux vector controlled by voltage vectors, as illustrated in Fig. 4. Thus, DTC can rapidly change θ and achieve a fast dynamic response.
The following is used to investigate the sensitivity of torque to θ (applying the ghost operator g): where ⊙ is the inner product operator. Considering constant flux operating conditions, the torque is proportional to sinθ, and the change rate of torque relative to θ is always positive. The torque change is realised by changing θ, and it is a positive feedback process. The feedback gain is proportional to the cosθ. Therefore, non-linear automatic gain control (AGC) is realised, which improves the torque response speed and restricts the positive feedback effect so as not to be uncontrolled. The motor is inherently en-stable and predominated by non-linear AGC positive feedback: the greater the absolute value of torque, the greater the absolute value of θ, and the weaker the positive feedback effect. Conversely, the smaller the absolute value of torque, the stronger the positive feedback effect.
The en-stable nature predominated by non-linear AGC positive feedback represented by inner product expression (13) is the main reason that DTC can improve the torque response speed. In addition, the inner product expression (13) clearly shows that if the amplitude of the flux declines, the motor will operate with large θ, and the positive feedback gain will be small. Thus, the torque response is sluggish during weak-magnetic operations.
To better describe and utilise en-stability characteristics predominated by non-linear AGC positive feedback, the AGC positive feedback gain is defined as a new variable R, and the sensitivity function of R is assessed.
The variable R is an expression based on the inner product of the stator flux and rotor flux vectors. The sensitivity of R to θ is equal to the negative torque expression. References [18][19] and [21][22][23] pointed out that the dynamics of flux amplitude are closely related to R. Presumably, the variable R can completely characterise the dynamics of flux amplitude, as shown in Fig. 4. From the sensitivity of R to θ, another characteristic can be observed: when the torque changes, the amplitude of flux has a self-stable nature predominated by nonlinear AGC negative feedback, and decoupled control of torque and flux can be realised. The larger the absolute value of the torque, the larger the negative feedback gain, and non-linear AGC negative feedback gain is expressed completely by the external product formula of the torque calculation.
The amplitude of the flux is predominated by the self-stable nature of the non-linear AGC negative feedback, and the voltage vector acts within the short time interval Δt. Thus, the change in flux amplitude can be ignored, and the change in the flux vector is mainly dependent on the change in θ. From the principles of DTC in Fig. 4, we have The following formula can be obtained in the short time interval Δt: The basic principles of DTC are well known: rotor flux dynamics are slower than those of stator flux, so it can be assumed that the rotor flux vector remains invariant during one sample time interval. Combined with the above analysis, in the short time interval Δt, the effect of the voltage vector on stator flux vector mainly changes θ, and the change in flux amplitude can be ignored. Therefore, we can draw the following conclusions: the essence of torque control is the control of the phase angle θ; the rotor flux vector remains invariant, and θ is controlled by torque T, so controlling reactive torque R is to control the flux amplitude, as shown in Fig. 4. The above analysis corresponds to the conclusion in [18] that the radial motion is relatively slow compared with the tangential motion. Therefore, the variable R and torque T can completely describe the action characteristics of voltage vectors to the torque and flux and fully express the characteristics of high-frequency signals of voltage vectors in the motor system. The expressions of R and T include but are not limited to the stator and rotor flux vectors. For example, using the inner-external products of stator flux and current vectors can obtain the same characterisation effect as the above analysis. To reduce the computational complexity of the proposed method, this paper selects inner-external products of the stator flux and current vectors for calculation in the following parts.
Overall, the variable R and torque T fully represent the dynamic characteristics of fast-varying voltage signals acting on the motor system. Using the variable R and T, we can fully describe the effects of voltage vectors on the torque and flux and realise decoupled control of the torque and flux. Moreover, the expressions of R and T indicate that the dimensions of R and T are identical and each possesses fast time-scale dynamics.
Therefore, this paper uses R and T with the same fast time scale and dimension as the references of the new cost function, which can simultaneously control torque and flux magnitude without the weighting factor, similar to predictive current control (PCC).
Where R* and T* are the reference values of variable R and torque T, respectively. With ψ s (k + 1) and i s (k + 1), variable R and torque T at the (k + 1) instant can be predicted as

Delay compensation
Many calculations are required in the digital implementation of MPC, which introduces a considerable time delay in actuation. This delay can diminish performance of the control system. To eliminate the delay in digital control, the cost function for minimisation is considered as where R(k + 2) and T(k + 2) are the predicted variable R and torque T at the (k + 2) instant, respectively. First, R* and T* at the (k + 2) instant should be determined. As the sampling frequency is much higher than the frequencies of the R and T references, R* and T* at the (k + 2) instant can be considered approximately equal to the present values, such that T* = T*(k + 2) and R* = R*(k + 2).
Second, we estimate the predicted stator flux ψ s (k + 1) and stator current i s (k + 1) according to (7) and (8) based on the voltage vector u s (k) determined in the previous control period; then, we calculate the rotor flux vector ψ r (k + 1) according to (9). Next, ψ s (k + 2) and i s (k + 2) can be simply predicted as ψ s (k + 2) = ψ s (k + 1) + T s u s (k + 1) − R s T s i s (k + 1) i s (k + 2) = 1 − T s τ σ × i s (k + 1) As the rotor mechanical constant is much greater than the sampling time, it is a general practice to assume ω(k) = ω(k + 1). Finally, the variable R and torque T at the (k + 2) instant can be predicted as Naturally, complicated prediction of rotor flux at the (k + 2) instant is eliminated.

Proposed control algorithm
The process of the proposed RT-MPC based on variable R and torque T characterisation can be summarised as follows (see Fig. 3): (1) Measure: Sample ω, i s , and dc-link voltage u dc .
(2) Apply: Set the optimal voltage vector found in the previous loop iteration.

Simulation study
To verify the performance of the proposed RT-MPC, it is simulated in a MATLAB/Simulink environment. Machine parameters are listed in Table 1. Fig. 5 shows simulated responses of MPTC at a speed of 500 r/min when the weighting factor λ ψ varies from 5 to 300. From top to bottom, the curves in Fig. 5 denote the speed, torque, stator current, and stator flux, respectively. When the weighting factor is set to 5, although the torque ripples are small, obvious ripples exist in the flux and current. When λ ψ increases from 5 to 30 (i.e. the ratio between rated torque and stator flux amplitude), a good balance between stator flux and torque ripples is achieved. When λ ψ increases further to 200 and 300, the flux ripples change negligibly, but the torque ripples and stator current harmonics are increase noticeably, which affects the steady operation of the system's speed loop. These results suggest that the weighting factor must be well designed to achieve satisfactory steady-state behaviour of torque and stator flux for MPTC. For the IM in this paper, λ ψ = 30 may be an appropriate solution for MPTC.
In this paper, the prediction of flux and torque is model-based. Mismatched parameters would affect control of torque and flux. To evaluate the influence of machine parameter variations on system performance, Fig. 6 presents the simulation results of RT-MPC at a moderate speed when the stator and rotor resistances are increased by 100%. The system works well, and the current is highly sinusoidal in shape, even if the stator and rotor resistances vary substantially from their nominal values. The simulated results show that the RT-MPC is robust against machine parameter variations.

Experimental results
To validate the performance of the proposed strategy, some experiments are carried out on a two-level inverter-fed IM drive   Fig. 7; the motor parameters are the same as in Table1. The sampling frequency of the MPTC in [6] and proposed RT-MPC is set to 10 kHz. The experimental results of MPTC with different weighting factors are shown in Fig. 8. The weighing factor exerts a significant influence on the torque and flux ripples. To achieve satisfactory performance, the weighting factor used in MPTC is set to 40 that is tuned based on a number of experimental tests.

Step speed transient characteristic
First, the dynamic behaviour of the RT-MPC is investigated and compared to MPTC.
Step speed transient behaviours of both methods are presented in Fig. 9. Initially, the motor is started at 150 r/min and then suddenly a step speed reference of 400 r/min is commanded. Similar to MPTC, the RT-MPC can track the reference speed quickly and accurately, and the rising speed times of both methods are comparable as indicated in Fig. 9.
As indicated by the above experimental tests, both methods achieve decoupled control of torque and flux during the dynamic process, and the RT-MPC has an equivalent dynamic response and similar steady-state behaviour to those in MPTC. Moreover, the RT-MPC does not require any weighting factor tuning work. In this sense, the proposed RT-MPC is simpler and more practical.

Step torque transient characteristics
Step torque transient characteristics of FOC, MPTC, and the proposed RT-MPC have been tested, as displayed in Fig. 10. A step torque reference of 10 Nm is commanded. The torque rise times of MPTC and the RT-MPC are quite fast (∼2.5 ms). On the contrary, FOC requires >10 ms, much slower than MPTC and RT-MPC. These results confirm that the proposed RT-MPC achieves a fast dynamic response, similar to other direct strategies, due to the absence of internal current control [14]. Furthermore, RT-MPC avoids use of the weighting factor and is thus more appealing.

Steady-state behaviour in low-speed operations
Next, the steady-state performance of MPTC and the RT-MPC is tested. Steady-state characteristics of both methods at a low speed with 5 Nm load are presented in Fig. 11. The torque and flux ripples of MPTC and RT-MPC are highly similar; the steady performance of RT-MPC is deemed comparable to that of MPTC with a fine-tuning weighting factor.

Investigation of average switching frequency
The average switching frequencies of both methods at different speeds and without load are presented in Fig. 12. The average switching frequencies of the MPTC and RT-MPC vary noticeably  over a wide speed range, whereas the average switching frequencies of the RT-MPC are slightly lower. This phenomenon may be due to the fixed weighting factor in MPTC throughout the entire speed range, which is generally not 'globally' optimal. By contrast, the RT-MPC uses T and its sensitivity function R with the same fast time scale and dimension, which enables the control performance to achieve a 'global' optimal more easily. Therefore, RT-MPC is more favourable for applications with low switching frequency.
Finally, we also test the execution time (prediction and actuation) of RT-MPC. Execution times of the conventional MPTC in [25] and the RT-MPC are very close: 27.68 versus 25.55 µs, respectively. Therefore, RT-MPC is similar to or even simpler than MPTC in computational complexity.

Conclusion
The proposed RT-MPC method is feasible and effective. Based on the variable R and torque T can fully represent flux and torque dynamic characteristics in the motor system, a hybrid MPC scheme has been proposed. By using T and its sensitivity function (dual function) R, the RT-MPC method achieves separation of the torque with fast time-scale dynamics and flux with slow time-scale dynamics (similar to FOC), eliminates the weighting factor (similar to PCC), and directly controls torque without the current loop (similar to DTC or MPTC). Thus, the presented RT-MPC of IM has the following features and advantages: The proposed RT-MPC algorithm has been verified through simulations and experiments. It exhibits good performance in steady-state and dynamic behaviour without any adjustment to the weighting factor, as is done in the MPTC scheme. The proposed method thus offers simple, reliable, and practical drive commissioning for MPTC applications.